Since \(2d_{q(.)} + 2 + \alpha + n \ge d_{p(.)}\) , as Theorem 8, given \(p_0\)  such that \(max \lbrace 1, p_{+} \rbrace  < p_0 < \frac{n}{\alpha }\) , we can decompose a distribution \(f \in H^{p(.)}(\mathbb {R}^{n}) \cap L^{p_0}(\mathbb {R}^{n})\)  as \(f = \sum _{j=1}^{\infty } \lambda _j a_j\) , where \(a_j\)  is an \((p(.), p_{0}, 2d_{q(.)} + 2 + \alpha + n)\) -atom supported on the cube \(Q_j\) , where the convergence is in \(H^{p(.)}(\mathbb {R}^{n})\)  and in \(L^{p_0}(\mathbb {R}^{n})\)  so it is enough to show that if \(a\)  is an \((p(.), p_{0}, 2d_{q(.)} + 2 + \alpha + n)\) -atom supported on the cube \(Q\)  centered at \(z\) , then \(cI_{\alpha }(a)\)  is a \((q(.), q_{0}, d_{q(.)})\) -molecule centered at a cube \(Q\)  for some fixed constant \(c > 0\)  independent of the atom \(a\) . Indeed, since \(f = \sum _{j=1}^{\infty } \lambda _j a_j\)  in \(L^{p_0}(\mathbb {R}^{n})\)  then \(I_{\alpha }f = \sum _{j=1}^{\infty } \lambda _{j} I_\alpha (a_{j})\)  in \(L^{\frac{n p_{0}}{n- \alpha p_0}}(\mathbb {R}^{n})\)  and thus \(I_{\alpha }f = \sum _{j=1}^{\infty } \lambda _{j} I_\alpha (a_{j})\)  in \(\mathcal {S}^{\prime }\) . Now from Theorem 5.2 in [1]} we obtain
\(\Vert  I_{\alpha }f \Vert _{H^{q(.)}} \lesssim \mathcal {A}(\lbrace  \lambda _j \rbrace , \lbrace  Q_j\rbrace , q(.))\) 
by Lemma 7 and Theorem 4.6 in [1]} we have
\( \lesssim \mathcal {A}(\lbrace  \lambda _j \rbrace , \lbrace  Q_j\rbrace , p(.)) \lesssim \Vert  f \Vert _{H^{p(.)}},\) 
for all \(f \in H^{p(.)} \cap L^{p_0}(\mathbb {R}^{n})\) , so the theorem follows from the density of \(H^{p(.)} \cap L^{p_0}(\mathbb {R}^{n})\)  in \(H^{p(.)}(\mathbb {R}^{n})\) .
Now we show that there exists \(c>0\)  such that \(cI_{\alpha }\left( a\right) \)  is a \(\left( q(.),q_{0},d_{q_{(.)}}\right) -\) molecule
\(1)\)  The Sobolev theorem and Lemma 4 give
\(\left\Vert I_{\alpha }\left( a\right) \right\Vert _{L^{q_{0}}(2\sqrt{n}Q)}\lesssim \left\Vert a\right\Vert _{L^{p_{0}}}\le \frac{\left|Q\right|^{\frac{1}{p_{0}}}}{\left\Vert \chi _{Q}\right\Vert _{p(.)}}\lesssim \frac{\left|Q\right|^{\frac{1}{q_{0}}}}{\left\Vert \chi _{Q}\right\Vert _{q(.)}}.\) 
\(2)\)  Denote \(d=2d_{q(.)}+2+\alpha +n.\)  As in the proof of Theorem 8 we
get, for \(x\)  outside \(2\sqrt{n}Q,\)  that
\(\left|I_{\alpha }\left( a\right) (x)\right|&\lesssim &\frac{l(Q)^{d+1}}{\left|x-z\right|^{n-\alpha +d+1}}\left\Vert a\right\Vert _{1}\lesssim \frac{1}{\left\Vert \chi _{Q}\right\Vert _{q(.)}}\left( \frac{l(Q)}{\left|x-z\right|}\right) ^{^{n-\alpha +d+1}} \\&=&\frac{1}{\left\Vert \chi _{Q}\right\Vert _{q(.)}}\left( \frac{l(Q)}{\left|x-z\right|}\right) ^{^{2n+2d_{q(.)}+3}} \\&\lesssim &\frac{1}{\left\Vert \chi _{Q}\right\Vert _{q(.)}}\left( \frac{l(Q)}{l(Q)+\left|x-z\right|}\right) ^{^{2n+2d_{q(.)}+3}} \\&=&\frac{1}{\left\Vert \chi _{Q}\right\Vert _{q(.)}}\left( 1+\frac{\left|x-z\right|}{l(Q)}\right) ^{^{-2n-2d_{q(.)}-3}},\) 
where the third inequality follows since \(x\)  outside \(2\sqrt{n}Q\)  implies \(l(Q)+\left|x-z\right|<2\left|x-z\right|.\)
\(3)\)  The moment condition was proved by Taibleson and Weiss in [3]}.
